Barcelona will never be able to compete with Real Madrid in their current state without Mbappe. There are many differences between the performances of these two teams, Real Madrid and Barcelona. Real Madrid is the best performer in every tournament as well as they are performing great in their league and their performance is consistent in contrast to Barcelona but they are not performing well in all tournaments and Barcelona performance is not consistent.
Real Madrid are signing Mbappe to further strengthen their squad so that they can beat the big teams in major tournaments. The main reason for Real Madrid signing Mbappe is not to compete with Barcelona or insult Barcelona, but to improve their overall performance.
Barcelona has not been a problem for Madrid this season. Even with the current squad I think Madrid is still capable of winning the La Liga title next season. So I think you are right that Madrid's goal in recruiting Mbappe is to improve the team's overall performance. With Mbappe, Real Madrid will be the best club in the world next season. And I'm sure they will be targeting a treble winner or even more next season. With Barcelona's current financial condition I think they will really not be able to compete with Madrid next season.
After all, it doesn't seem to be something strange or something new about Real Madrid always managing to bring in high-quality players. Because after all, it seems that Real Madrid is also a destination for quality players, so it is not surprising that there are always quality players who accept Real Madrid's offer in every new era that occurs. By having such a history, then of course Real Madrid has always managed to also maintain its tradition by having star players in the Real Madrid squad. Of course, with these factors, it is not surprising that Real Madrid is not only good in La Liga, but Real Madrid is also the most team that won the trophy in the Champion League and thus, Real Madrid deserves to be called the king of Europe.